Positive solution for a class of nonlinear fourth-order boundary value problem

1.   Introduction

Nonlinear mathematical models ^[1,2] were widely used in many fields. In particular, boundary value problems of nonlinear differential equations have received extensive attention and have been intensively studied in the past thirty years, see ^[3,4]. We point out that boundary value problems for second order differential equations, see, for example ^[5,6,7,8,9] and the references therein. While studies about boundary value problems of nonlinear fourth-order differential equations are much more less. One of the earliest papers about boundary value problems of nonlinear fourth-order differential operator is ^[10] from R. Ma and H. Wang, there they concerned the following problem

with boundary condition

or

By the fixed point theorem in cone, they proved the existence of positive solutions under the conditions that $f$ is either superlinear or sublinear. In another paper ^[11], the author obtained the positive solution of the following problem

by the fixed point theorem in cone. R.Vrabel ^[12] studied the upper solution and lower solution of the problem

There are many other papers we will not list but we find that they have a common point, that is, the fourth-order differential operators they dealt with can be resolved into composition of two second-order positive linear operators. And therefore, the corresponding Green's function for fourth-order linear operator is the form of the product of two Green's functions for second-order linear operators.

In a recently paper ^[13], Drábet discussed the existence of positive solutions for the following fourth-order linear problem

Obviously, the fourth-order differential operator can not be resolution into composition of two second-order positive linear operators. For more results on nonlinear fourth-order differential operator problems we can refer to ^[14,15].

Based on the above literature inspiration. We now consider the fourth-order nonlinear equation with Dirichlet boundary conditions

where $c(x), a(x)$ satisfy some conditions that we will give bellow, especially, $a(x)$ may change signs.

We make the following assumptions throughout the paper:

(A1) $-\pi^{4} < c(x) < c_{0}$,

(A2) $f:\mathbb{R^{+}}\rightarrow \mathbb{R}$ is continuous, and $f(0) > 0$,

(A3) $a:[0, 1]\rightarrow \mathbb{R}$ is continuous with $a(x)\not\equiv 0$, and there exists a constant $K > 0$ such that

for every $x\in (0, 1)$, where $a^+$(resp. $a^-$) is the positive (resp. negative) part of $a$, $c_{0}$ is the constant given in ^[13], and $G(x, y)$ is the Green's function of $L_{c}$ with boundary conditions (1.2).

Our main result is as follows:

Theorem 1.1. Let (A1)–(A3) hold. Then there exists a positive number $\lambda^*$ such that (1.1) and (1.2) have a positive solution for $0 < \lambda < \lambda^*$.

Remark 1.1. Since the fourth-order differential operator can not be resolution into composition of two second-order positive linear operators, as a result, the Green's function have no explicit expression. So the method or technic used in ^[10,11,12] does not work. To deal with the new case and the difficult it brings, we are inspired by the method to second-order elliptic boundary value problems in ^[8], and the result that the fourth-order operator $u^{(4)}+c(x)u$ is strictly inverse positive in ^[13,16]. Thanks to the existence and its properties of the Green's function given in ^[17,18,19], we obtain the existence of a positive solution to the problems (1.1) and (1.2).

2.   Preliminaries

In this section we present two important lemmas. The main method we use is the fixed point theorem of Leray-Schauder type. We refer interested readers to the literature ^[20,21].

Set

and let the linear operator $L_{c}:W\rightarrow C([0, 1])$ defined by

Then the boundary value problems (1.1) and (1.2) are equivalent to the operator equation

Lemma 2.1. Let (A1) hold. Then $L_{c}$ is strictly inverse positive, and therefore it has a positive Green's function.

Proof. $L_{c}$ is strictly inverse positive, we can refer to ^[13,16] and the reference therein. From the definition of $L_{c}$ is strictly inverse positive there and the well-known truth that

is equivalent to

we can get the positiveness of the Green's function $G(x, y)$ immediately.

Lemma 2.2. Let (A1)–(A3) hold, and let $0 < \delta < 1$. Then there exists a positive number $\bar{\lambda}$ such that, for $0 < \lambda < \bar{\lambda}$, the problem

has a positive solution $\tilde {u}_\lambda$ with $|\tilde{u}_\lambda|\rightarrow 0$ as $\lambda\rightarrow 0$, and

where $p(x) = \int_0^1G(x, y)a^+(y)dy$.

Proof. It follows from Lemma 2.1 that $L_{c}$ is strictly inverse positive, and therefore it has a positive Green's function $G(x, y)$. For each $u\in C([0, 1])$, let

Then the fixed points of $A$ are solutions of problems (2.1) and (2.2). We now verify the condition of Leray-Schauder fixed point theorem to show that $A$ has a fixed point for $\lambda$ small.

Firstly, $A: C([0, 1])\rightarrow C([0, 1])$ is completely continuous by the assumptions and Arzela-Ascoli theorem.

Secondly, we find a bounded open set $\Omega$ with $0\in\Omega$ in $C([0, 1])$, such that for $u\in C(\bar{\Omega})$ and $\theta\in (0, 1)$, if $u = \theta Au$, then $u\bar\in \partial\Omega$.

By $(A2)$, the function $g(s) = \frac{f(s)}{f(0)}$ is continuous and $g(0) = 1$, since $0 < \delta < 1$, we can choose $\varepsilon > 0$ such that

Also we have

where $\tilde{f}(t) = \max_{0\leq s\leq t}f(s)$, and $|\cdot|_0$ is the usual norm in $C([0, 1])$.

Suppose $\lambda < \frac{1}{2|p|_0\tilde{f}(\varepsilon)} = : \bar{\lambda}$, then there exists a $A_\lambda\in (0, \varepsilon)$ such that

Let $\Omega = \{u\in C([0, 1]):\; |u|_0 < A_\lambda\}$ and $\theta\in(0, 1)$ such that $u = \theta Au$. Then we have

or

So $u\neq A_\lambda$, which means $u\bar\in \partial\Omega$.

By the Leray-Schauder fixed point theorem, $A$ has a fixed point $\tilde{u_\lambda}$ in $\Omega$ for $0 < \lambda < \bar{\lambda}$, that is, problems (2.1) and (2.2) have a positive solution $\tilde{u}_\lambda$ with $\tilde{u}_\lambda\leq A_\lambda < \varepsilon$. Notice that $A_\lambda\rightarrow 0$ as $\lambda\rightarrow 0$, $|\tilde{u}_\lambda|\rightarrow 0$ as $\lambda\rightarrow 0$ and

The proof is completed.

3.   Proof of Theorem 1.1

Proof of Theorem 1.1. Let $q(x) = \int_0^1G(x, y)a^-(y)dy$, recall that $p(x) = \int_0^1G(x, y)a^+(y)dy$. By $(A2)$,

From the proof of Lemma 2.2 that $g(0) = 1$, there is a $\alpha\in(0, 1)$ and we can choose $1 < \sigma < K$, such that $f(s) < \sigma f(0)$, and $\gamma = \frac{\sigma}{K}\in(0, 1)$, then we have

for $s\in[0, \alpha], \; \; x\in(0, 1)$. Fix $\delta\in(0, 1)$ and let $\lambda^* > 0$ be such that

for $0 < \lambda < \lambda^*$, where $\tilde{u}_\lambda$ is the solution of (2.1) and (2.2) given by Lemma 2.2, and

for $s, \; t\in [-\alpha, \alpha]$ with $|s-t|\leq\lambda^*\delta f(0)|p|_0$.

Let $0 < \lambda < \lambda^*$, we look for a solution $u_\lambda = \tilde{u}_\lambda+v_\lambda$. Since $\tilde{u}_\lambda$ is the solution of (2.1) and (2.2), then $v_\lambda$ solves

For each $w\in C([0, 1])$, let $v = Aw$ be the solution of

where the operator $A$ is as in Lemma 2.2, we have

and $A$ is completely continuous.

Let

if $v\in C(\bar{\Omega'})$ and $\theta\in (0, 1)$, such that $v = \theta Av$, that is

we are going to show that

Suppose the contrary that $|v|_0 = \lambda\delta f(0)|p|_0$. Then by (3.2) and (3.3), we get

and

together with (3.1) implies that

and

a contradiction.

By the Leray-Schauder fixed point theorem, A has a fixed point $v_\lambda$ in $\bar{\Omega'}$ with $|v_\lambda|_0\leq\lambda\delta f(0)|p|_0$. Hence $v_\lambda$ satisfies (3.4), and using Lemma 2.2, we obtain

We have proved that $u_\lambda$ is a positive solution of (1.1) and (1.2).

4.   Conclusions

In this paper, we mainly study the existence of solutions to a class of nonlinear fourth-order Dilrichlet boundary value problems through Leray-Schauder's fixed point theorem, and show the asymptotic behavior of the solution as $\lambda$ changes. In the future, we can try to construct such solutions, give the properties of the solutions, or study numerical solutions for such problems.

Acknowledgments

The authors would like to thank the referees and editors for providing very helpful comments and suggestions. This work is supported by Natural Science Foundation of Gansu Province (No. 21JR1RA317) and Young Doctor Fund project of Gansu Province (No. 2022QB-173).

Conflict of interest

The authors declare no conflicts of interest in this paper.